On the dynamics of prebiotic evolution

Transcription

1 On the dynamics of prebiotic evolution Kelley Harris Advisors: Irene Chen (Systems Biology) and Clifford Taubes (Mathematics) April 17, Introduction The question of how life arose from non-living matter is the focal mystery where chemistry and molecular biology converge with evolutionary biology But although it exemplifies the scientific challenge of explaining how and why complex phenomena appear in nature, it defies standard methods of scientific inquiry The first life to assemble abiotically left no fossils, and scientists are unlikely to prove anything definite about its identity Instead, origins of life researchers look for conditions under which non-living material assumes certain properties of living material, trying to bridge the chasmal gap that separates living and nonliving chemistry today In 1828, Friedrich Wohler s urea synthesis proved that organic molecules can be obtained from inorganic ones via ordinary chemical reactions [36], debunking the 17th century belief that organic materials owe their heat-sensitivity to a vital energy that is absent from inorganic material [37] Wohler s work ignited a science of organic chemistry that has used simple physics to explain most unique properties of biomolecules; in particular, chemists have found many chemical conditions that promote predictable interconversions between organic and inorganic matter In 1959, Stanley Miller proved that such conditions need not be highly artificial, synthesizing DNA and protein building blocks within an apparatus that simulated a lightening storm over a pool of common sludge [27] James Ferris later proved that charged clay surfaces and drying conditions are enough to promote the polymerization of complex RNA chains [13] Such work has effectively demystified the chemical split between life and non-life; although we will never know with certainty how the chemical components of life first appeared on Earth, their presence can be explained in many plausible ways However, there are no equally satisfying explanations for the advent of self-replication and metabolism, behaviors that biochemicals only exhibit when organized into complex systems By creating a scientific landscape where the difference between synthetic organic material and natural organic material is organizational complexity, 19th and 20th century biochemists have turned the origins of life question from a chemical synthesis problem into an information-theoretic problem The supposed manifestation of 17th century vital force in heat sensitivity was spot on in that a protein fails to reconstitute after heating because the information content of its three-dimensional fold has been destroyed When a protein cannot assume its bio-active, water-soluble native configuration without help from the cellular machinery that its own catalytic activity helps maintain, we are faced with the sort of chicken-or-egg dilemma that creates so much of the mystique surrounding the origins of biological systems But just as paleontologists know that eggs preceded chickens by millions of years, modern biochemists know that there exist certain proteins that are capable of reconstituting after being heated because they assume their native folds spontaneously Furthermore, it is well understood via the theory of energy landscapes why certain proteins fold spontaneously and others do not [20] The information contained in a spontaneously folding protein sequence can propagate itself if it encodes a structure that catalyzes its own replication, as was demonstrated by the 1

2 synthesis of a self-replicating peptide in 1996 [26], and the popular RNA World hypothesis postulates that such self-replicating catalysts were the first carriers of genetic information This hypothesis favors self-replicating RNA molecules over self-replicating proteins because RNA catalysts tend to be simpler than protein catalysts and replicate in a more uniform and straightforward way However, no protein or nucleic acid polymer has ever been replicated in the laboratory without the help of enzymes derived from cells Chemists have spent decades trying to synthesize a ribozyme that catalyzes RNA-dependent RNA polymerization faithfully enough to replicate itself, and such an RNA replicase has proved elusive In 2001, Johnston, et al reported the synthesis of a ribozyme that is capable of RNA-dependent RNA polymerization, but found that it disintegrated after less than 20 nucleotide additions under the harsh chemical conditions that were required to maintain catalytic activity [23] Although it is not known how many potential replicases are crippled by catalytic dependence on degradative chemicals, it is possible to mathematically derive some universal constraints on successful molecular replicators The earliest of these constraints emerge from the quasispecies model that was developed in the 1970s by theoretical chemists Manfred Eigen and Peter Schuster [5, 6, 7, 8, 9, 10] In addition to revealing constraints on the viability of an early molecular replicator, quasispecies theory describes viral population dynamics with remarkable accuracy [29] Eigen and Schuster also discuss the origin of translation, arguing that an RNA world must have given way to an RNA-and-protein world via a chemical reaction they call a hypercycle [7, 8, 9] However, they say little about chemical systems that predated the advent of virus-like replicators, and it was not until 2008 that Nowak and Ohtsuki modeled the emergence of virus-like replicators from polymers that did not replicate, assembling without templates by a process like clay surface catalysis 1 [28] When viewed end-to-end, these three bodies of theory constitute a scenario for the transformation of disorganized biomolecules into dynamic populations of self-replicating polymers that eventually become capable of protein synthesis My review article will present the theory of this remarkable transformation in the order that it was published, beginning with the quasispecies in Sections 2 and 3, moving to the hypercycle in Section 4, and ending with Nowak and Ohtsuki s prevolutionary dynamics in Section 5 Such mathematical models form a cornerstone of origins of life research because even experimental results about the origins of life depend strongly on models of neverobserved phenomena Although a successful experiment can never prove that a certain event actually took place in the prebiotic world, a mathematical result is better suited to proving that a certain chemical process can never have helped bring about the origin of life In addition to providing truly absolute knowledge about the history of life on earth, mathematical results can guide experimentalists toward types of chemical reactions that are likely to display life-like behavior and steer them away from reactions that will never display such behavior Although it is far from a historical account of early evolution, the theory discussed here is central to our current understanding of the ontogeny of biochemistry 2 A dynamical model of self-replication This paper will model the emergence of life-like properties in systems of large molecules whose behavior is best described by the theory of chemical kinetics Specifically, we consider a solution populated by n chemical species x 1,, x n whose players interconvert via chemical reactions of the form a 1 x a n x n b 1 x b n x n 1 In [28], Nowak and Ohtsuki treat polymer extension and replication as abstract processes, preferring not to situate their evolutionary dynamics within a particular chemical framework They would not identify their prelife landscape with anything as concrete as the clay surface polymerization experiments reported in [13], and the viability of their models is not bound up with the viability of clay surface polymerization as a precursor to replication However, my goal is to piece together an origins of life scenario that is as concrete as possible, and so I will take the liberty of identifying Nowak and Ohtsuki s evolutionary processes with illustrative chemical processes Page 2

3 for some integers {a i }, {b i } [5] We will let [A] denote the concentration of species A in solution, and let k denote the rate constant of the forward reaction, such that the reactants turn into products at the rate of k[x1 ] a1 [x n ] an moles per liter per second Similarly, we will let k denote the rate constant of the reverse reaction, meaning that products turn into reactants at the rate of k [x1 ] b1 [x n ] bn moles per liter per second We will say that a system is at equilibrium whenever the set of concentrations [x 1 ],, [x n ] is a solution to the equation k[x1 ] a1 [x n ] an = k [x 1 ] b1 [x n ] bn, such that the forward and reverse reactions happen at equal rates and the concentrations [x 1 ],, [x n ] remain constant over time A system is far to the left of equilibrium when the chemical concentrations render the reverse reaction much slower than the forward reaction, such that the overall rates of change d[x i ]/dt coincide with the forward reaction rate Similarly, a system is far to the right of equilibrium when the forward reaction is much slower than the reverse reaction A reaction is autocatalytic whenever there exists x i for which b i > a i > 0, in which case x i is called a self-replicating species When a system subject to this autocatalytic reaction is far to the left of equilibrium, the concentration of the self-replicating species x i will increase at a rate of (b i a i ) k[x 1 ] a1 [x n ] an moles per liter per second But as a system approaches equilibrium, the rate of change (b i a i )( k[x 1 ] a1 [x n ] an k [x 1 ] b1 [x n ] bn ) of [x i ] approaches zero We conclude that autocatalysis can lead to the self-replicating behavior we associate with life, but only in systems that lie far from chemical equilibrium [5] All isolated chemical reactions fall toward equilibrium with the passage of time; 2 a truly living entity must be capable of metabolism, which pumps energy into the system in a way that keeps autocatalytic processes from nearing equilibrium Many biologists believe that replication preceded metabolism in that the earliest self-replicating entities did nothing to stave off chemical equilibrium and were mining an existing chemical gradient that they could not maintain on their own Although a few scientists believe that metabolism actually preceded replication [30], this paper will focus on replication-first scenarios in which the first precursors to life were passively autocatalytic 3 The next section will construct a 2 A few chemical reactions never stabilize near equilibrium, but instead oscillate along a limit cycle of nonequilibrium states The first chemical oscillator reported in the literature is a mixture of potassium bromate, cerium (IV) sulfate, propanedioic acid, and sulfuric acid; as the cerium ions oscillate between an oxidized state and a reduced state, a yellow color repeatedly appears and disappears [1] Some chemists consider this Belousov-Zhabotinsky reaction to be a good theoretical model for nonequilibrium biological dynamics, but it is hard to imagine how a chemical oscillator could beget the stable molecular replicators that focus discussions about the origins of life By definition, living things must propagate themselves, and it is impossible for an oscillator trapped within a limit cycle to beget copies of itself without disassembling these copies to begin the cycle again 3 In a seminal paper on the dynamics of early replicators [7], Eigen lists metabolism, self-reproduction, and mutation as the three chemical prerequisites for evolution by natural selection However, he defines metabolism to be the continuous formation and degradation of chemical structures that occurs in all nonequilibrated systems It is more standard to define metabolism to be a mechanism for maintaining chemical non-equilibrium; we classify Eigen s model system as pre-metabolic because it does not include a mechanism for maintaining chemical gradients and will only be subject to natural selection as long as external forces keep it far from equilibrium Page 3

4 model population of such passive autocatalysts, and will prove several results about the evolution of this population This model includes no provisions for population structure or other interactions among the replicators, and thus represents the simplest possible model of replicator dynamics 3 Primitive replicators and quasispecies theory The work that follows will present Eigen s basic assumptions about a model population of autocatalytic entities and prove several important results that he introduced in [7] To begin, we will let x 1,, x n denote self-replicating polymers and write down n differential equations describing the rates of change of their concentrations as they react with activated monomers to make copies of themselves When we write out these differential equations explicitly, we will omit concentration brackets and let each variable x i denote the concentration of a molecular species We will assume that the reaction conditions remain constant by virtue of a constant influx of activated monomers balanced by an outflux of polymers and their degradation products In addition, we will assume that the rates at which x i forms and degrades depend linearly on [x i ], as is the case when one polynucleotide catalyzes the synthesis of one complementary strand by acting as a template In a system that lies close to equilibrium, chemical reactions tend to be reversible, meaning that the conversion A + B C is a microscopic reversal of the conversion C A + B In contrast, systems far from equilibrium are usually dominated by irreversible reactions where the conversions A+B C and C A+B proceed by completely different mechanisms Since self-replicating molecules must inhabit chemical systems that lie far from equilibrium, it follows that their formation and degradation will proceed by distinct pathways with different rate-dependencies For example, the rate A i x i at which x i templates the production of new polymers generally depends much more strongly on catalytic activity than does the rate D i x i at which x i is degraded Another source of the polymer x i is the imperfect replication of related polymers x j ; when the sequences x i and x j differ by only a few monomer insertions, deletions, and substitutions, x j will catalyze the production of x i at the rate w ij x j Conversely, there is a nonzero probability 1 Q i that x i will replicate itself imperfectly and produce some mutant polymer Finally, a solution outflux will carry away x i at the rate Φ i (x) If we neglect all other factors that affect the concentration of x i, we conclude that a set of differential equations of the form x i = (A i Q i D i )x i + w ik x k Φ i (x) (1) k i describes the evolution of the system at hand In addition to catalyzing its own replication at a rate of A i Q i x i, x i catalyzes the production of mutant polymers at a rate of A i (1 Q i )x i When {x 1,, x n } is the complete set of polymers that inter-mutate within the chemical system at hand, all of the mutants produced by the imperfect replication of x i are members of this set In this case, the individual mutant production rates w ji must sum to the net production of mutants A i (1 Q i ) that occurs during the replication of x i : A i (1 Q i ) = j i In test tube evolution experiments, it is common to regulate the flow rates Φ i so that reaction conditions remain constant and reproducible Specifically, it is convenient to ensure that the total replicator concentration remains constant: Φ i = A i x i w ji D i x i (2) Page 4

5 We will assume that the Φ i s have this property so that our system of differential equations will model a snapshot of evolution where conditions are approximately constant and selective effects can be observed most clearly We constrain the individual Φ i s so that polymers x i leave the system in proportion to their representation in the pool: ( n ) x i Φ i = Φ k n k=1 x (3) k k=1 In order for the n equations of the form (1) to furnish a physically meaningful model of evolution, they must ensure that positive concentration variables never become negative as the system evolves We will always consider initial conditions for which every x i is nonnegative, and if we suppose for the sake of contradiction that some x i is negative at time t, then there exists a time t 0 < t at which x i = 0 and x i < 0 Since Φ i is regulated according to (3), it vanishes at time t 0, meaning that x i (t 0 ) = w ik x k (t 0 ) k i Without loss of generality, we can assume that no x k becomes negative before x i does Therefore, since each w ik is nonnegative, we have a contradiction, and we can be assured that the x i population variables remain nonnegative indefinitely If we assume that outflow is regulated according to (2) and (3) and let E i = A i D i denote the excess productivity of the replicator x i, then we can let n E(t) = E ix i n x i denote the average excess productivity as a function of time This lets us write the replicator equations in a homogeneous form: x i = (A i Q i D i E(t))x i + w ik x k k i When the fitness values W i = A i Q i D i and the mutation probabilities w ik are constant with respect to time, this system of differential equations is solvable An exact solution is reported in [33], and good approximate solutions can be obtained using perturbation theory The replicator equations highlight the fact that no primitive autocatalyst could realistically drive all other autocatalysts to extinction Even a master sequence x i whose fitness value W i is much higher than all competing W j s continuously populates a range of nearby mutants, except in the chemically unrealistic case Q i = 1 This process produces a distribution of replicators centered at x i, and Eigen presents a mathematical argument that the replicator mixture will stabilize over time, converging to an eigenvector of the mutation matrix W whose diagonal entries are the W i s and whose off-diagonal entries are the w ij s If we let x i denote the n-entry vector whose jth entry is x i δ ij, then by construction, the replicator population will converge to the stationary state lim k W k x i, which is an eigenvector of W by the theory of finite-state Markov chains 4 It is thus more appropriate to say that natural selection acts on eigenvectors than on individual replicator species; for this reason, Eigen refers to the eigenvectors of W as quasispecies His argument implicitly assumes that the spectral radius of W is an eigenvalue of W and that the corresponding eigenvector (c 1 c n ) T has nonnegative entries, such that the linear combination c 1 x 1 + c n x n is a physically meaningful sequence distribution to which the chemical solution makeup can 4 The convergence of a replicator population to a steady state concentration vector looks uncomfortably similar to the onset of chemical equilibrium, and we will even refer to such a steady state as an equilibrium point However, a steady-state population lies far from chemical equilibrium as long as it depends upon a constant influx of energy-rich molecules such as activated nucleotides, which is the case for every replicator population at equilibrium that we will encounter in this paper Page 5

6 converge We can justify both of these assumptions using the following theorem, which is a standard linear algebra result proved in references like [19]: Theorem 1 (Perron-Frobenius Theorem, [19]) If A = (a ij ) is a real n n matrix with positive entries a ij > 0 and eigenvalues λ 1,, λ n, then the following statements hold: 1 There is a unique positive real eigenvalue λ i for which max{ λ 1,, λ n } = λ i 2 λ i is a simple root of the characteristic polynomial of A 3 λ i is associated with an eigenvector that has strictly positive entries Conversely, there is a unique nonnegative eigenvector (v 1,, v n ) of A for which v v n = 1 4 The matrix entries obey the inequality min a jk λ i max a jk j j k Assuming that the x i s are all potentially viable replicators for which A i Q i D i 0, such that each would prosper in the absence of competitors, W has nonnegative entries In addition, the mechanics of nucleic acid replication generally ensure that if there is a nonzero probability that x i will give rise to x j after n generations, then there is a nonzero probability that x i will give rise to x j after one generation With this justification, it will be convenient to assume that W is a block diagonal matrix for which each block W i has strictly positive entries When W i and W j are entries in separate blocks, the corresponding replicators x i and x j do not catalyze one another s replication, but only interact by competing for resources If we let λ i1,, λ im be the m eigenvalues of the block W i, then the Perron-Frobenius theorem guarantees that there exists a real λ ik for which λ ij < λ ik for all j k and that the coefficients of the eigenvector y ik = c ik 1x c ik nx n are nonnegative We also define scalar concentration variables y 1,, y n such that when y i = c i1 x c in x n, we have y i = c i1 x c in x n When the c ij s are nonnegative, the quasispecies has physical meaning as a pool of related polymers that coexist in specific ratios In addition, when a physically meaningful quasispecies y k is associated with a positive real eigenvalue λ k, we can see that y k produces offspring at a rate of λ k y k, such that the size of λ k dictates the fitness of y k Thus, although it will be mathematically expedient to model the system s evolution as a competition among quasispecies that do not all have physical meaning, the fittest quasispecies will always be physically meaningful entities whose success can be experimentally measured These quasispecies will behave as if they are competing against less fit quasispecies that do not have the same physical meaning, but these imaginary quasispecies will disappear with time Since the set of real n n matrices that are not diagonalizable over C is the complement of the measure-zero hypersurface in R n n of matrices with the property that the discriminant of the characteristic polynomial vanishes, we can perturb W infinitesimally if necessary and assume that it is diagonalizable over C, such that the eigenvectors y 1,, y n are linearly independent Therefore, there exist coefficients c 1,, c n for which n x i = n c iy i, and if necessary, we can infinitesimally perturb W to ensure that the c i s are all strictly positive After doing so, we can choose c 1 y 1,, c n y n as our representative quasispecies eigenvectors and assume that n x i = n y i Proposition 2 The average excess productivity E(t) is invariant under a change to quasispecies coordinates, with k E(t) = E 1x E n x n x x n = λ 1y λ n y n y y n Page 6

7 Proof Let n j=1 (Wx i) j be the sum of the entries of the vector Wx i, with respect to the basis where (x i ) j = x i δ ij By construction, we can see that (Wx i ) j = (A i Q i D i + w ik )x i = E i x i k i j=1 We now have E(t) = n n j=1 (Wx i) j n x i Since the vectors y 1,, y n span replicator concentration space, there exist coefficients k i1,, k in for which x i = k i1 y k in y n Furthermore, since n x i = n y i, it must be true that n k ij = 1 Therefore, n n j=1 E(t) = (Wx n n i) j j=1 = (W(k i1y k in y n )) j x x n y y n n n j=1 l=1 = (k n n i1 + + k in ) (Wy l ) j j=1 l=1 = λ l (y l ) j y y n y y n Letting y i = c i1 x c in x n as before, we now have n j=1 n l=1 λ n n l (y l ) j j=1 l=1 = λ l(c l1 x c ln x n ) j y y n y y n n l=1 = λ l(c l1 x c ln x n ) = λ 1y λ n y n y y n y y n In terms of this coordinate-invariant quantity, the equations describing the evolution of the system take the form y i = (λ i E(t))y i We will now state and prove a proposition implying that our system of replicators evolves toward a state populated entirely by the fittest quasispecies: Proposition 3 Let W be a mutation matrix with eigenvalues λ 1,, λ n and quasispecies eigenvectors y 1,, y n As the replicator system described by W evolves with time, we can show that lim t y i = 0 whenever λ i < max k n λ k Proof We can describe the evolution of this system over a large time interval T up to an arbitrary level of precision by dividing T into m small time intervals [T 0, T 1 ],, [T m 1, T m ] such that E(T i + t) t<ti+1 T i E(T i ) can be regarded as a constant and y i t<ti+1 T i can be regarded as a simple exponential function: y i t<ti+1 T i C exp ((λ i E(T i ))t) Page 7

8 If λ 1 = = λ n, then there is nothing to prove Otherwise, there exists i for which λ i < E(T 1 ) When λ i < E(T j ), y i will decline exponentially over the interval [T j 1, T j ]; similarly, y i will increase exponentially over T j whenever λ i > E(T j ) Therefore, E(T i+1 ) = k λ ky k (T i+1 ) k y k(t i+1 ) > k λ ky k (T i ) k y k(t i ) = E(T i) If we let S i denote the set of i [n] = {1,, n} for which λ i > E(T i ), then we can see that lim t y i = 0 for all i [n] \ S i Therefore, we need only suppose that λ l < max k n λ k and show that there exists T j for which λ l < E(T j ) We have already shown that the set S i has the property that lim E(t) = t k S i λ k y k k S i y k Therefore, either λ j = λ k for all j, k S i, or there exist λ j S i and k > i for which E(T k ) > λ j We deduce that S k S i, such that lim E(t) = j S k λ j y j t j S k y j Therefore, the set m S i contains exactly those λ i that equal max j n λ j, and all other λ k have the property that lim t y k = 0 We can think of a dominant quasispecies as a collection of mutant sequences that cluster around a master sequence whose fitness is maximal Theory and experiment show that the master sequence rarely makes up more than a few percent of the replicator population [7, 29]; although each individual mutant is much less abundant than the master sequence, the vast number of possible mutants lets them dominate the population But despite the relatively low abundance of master copies, the fitness of a quasispecies depends much more upon the fitness of the master than upon the fitness of the mutants Using second-order perturbation theory, Eigen and Schuster compute that the largest eigenvalue λ m of a mutation matrix without fitness level degeneracies is approximately W m + k m w km w mk W m W k, where W m is the fitness of the master sequence [7] For polymers that replicate with low error rates, the w ij terms will be much smaller than W m, and so the fitness of the master quasispecies will be very close to the fitness of its master sequence However, high replication error rates can lead to the success of quasispecies that are much fitter than their master sequences We have seen that a quasispecies centered at x m will proliferate if and only if x m produces more offspring than its competitors produce on average, such that k m A m D m > (A k D k )x k k m x (4) k However, it is possible for the concentration of x m to decrease at the same time that a quasispecies centered at x m proliferates As mentioned earlier, the number of different mutants that can be produced by imperfect replication of x k is so great that the abundance of any particular mutant in the resulting pool is very small Accordingly, the amount of x m that is produced by back-mutation is negligible, and so the frequency of perfect copies of Page 8

9 x m will only increase with time if x m makes perfect copies of itself at a rate that exceeds the average productivity of competing sequences, such that k m A m Q m D m > (A k D k )x k k m x (5) k In a system where the fittest quasispecies is centered around a sequence satisfying (4) but not (5), we observe a phenomenon known as the error catastrophe When each individual sequence frequency x i approaches zero as time approaches infinity, the polymers in solution distribute themselves uniformly across sequence space In reality, a finite sequence population will sample sequence space stochastically, evolving by random drift as every sequence that appears inevitably dies out [10] Inequality (5) dictates a sharp error threshold that is usually represented as a minimum replication accuracy that is required to propagate a master sequence of a given length and fitness It is standard to assume that the replicators are assembled by adding monomers one by one to a growing chain, and that the probability of incorporating the wrong monomer during any given assembly step is q 5 If the replicators x 1,, x n are polymer chains of length ν, then Q 1 = = Q n = q ν Therefore, we can let σ m = D m + A m P k m (A k D k )x P k k m x k denote a superiority parameter for the dominant species x m catastrophe will happen if and only if and observe that an error ν > log σ m 1 q Unlike many results in the theory of prebiotic evolution, the existence of an error threshold implies many testable claims about modern organisms Experimentalists have created error catastrophes in vitro by sabotaging error-correcting mechanisms that maintain viral mutation rates below the error threshold [7] In addition, they have measured a general inverse correlation between a genome s size and the effectiveness of its error-correcting machinery (see Figure 1) An RNA-dependent RNA polymerase like viral Qβ replicase is accurate enough to preserve the information contained in a genome about 10 4 nucleotides long, which is the maximum length of a single-stranded RNA virus genome Double-stranded genomes allow for error checks that reduce the mutation rate by as much as three orders of magnitude; it has been observed that double-stranded virus genomes can be 10 5 nucleotides long, while bacterial genomes can be 10 7 nucleotides long [7] Eukaryotic organisms use still more complex error-correction mechanisms, some made possible by diploidy, to achieve even smaller error rates that make larger genomes possible Although the error threshold furnishes a beautifully nonobvious explanation for the inverse correlation between genome size and mutation rate, it must be pointed out a genome encoding complex error-correcting enzymes must be large enough to encode such enzymes Biologists estimate that no genome with fewer than 10 4 nucleotides could encode a system capable of faithfully translating DNA sequences into protein sequences [7]; however, we noted earlier that 10 4 nucleotides is the maximum length that single-stranded RNA genomes can achieve It is reasonable to assume that modern RNA viruses have evolved polymerases 5 Although both of these assumptions are standard, they represent oversimplifications In modern DNA and RNA synthesis, purine purine and pyrimidine pyrimidine substitutions are much more common than purine pyrimidine substitutions [7] It is also possible that early template-directed polymerization was less organized than modern nucleic acid synthesis and did not proceed by the sequential addition of monomers Unpublished computer simulation data suggest that more chaotic systems also exhibit error thresholds; however, it is easiest to put a numerical value on the error threshold when we make the stated assumptions about replicator assembly mechanics Page 9

10 Figure 1: This graph, taken from a recent paper reporting that 399-nucleotide viroid genome has the exceptionally high error rate of 1/400 [16], illustrates the fact that the error threshold pressures large genomes to evolve high-fidelity replication mechanisms that are as accurate as is chemically possible, and such polymerases could not evolve without the help of a translation mechanism that intertwines the fitness of a protein enzyme with the fitness of the polynucleotide genome it replicates We conclude that independent polynucleotide replicators that have not yet evolved to encode translation machinery can never grow long enough to encode such machinery This hypothetical barrier for molecular evolution is known as Eigen s paradox Scientists have proposed several solutions to Eigen s paradox, the most popular being the hypothesis that RNA can catalyze high-fidelity RNA replication An RNA replicase capable of reproducing sequences of 10 4 nucleotides could have maintained pools of replicators that eventually gave rise to protein translation machinery, resolving the chicken-or-egg dilemma that we described earlier But although molecular biologists have spent decades searching for such a molecule and have created ribozymes that catalyze template-directed RNA polymerization [23], none of these ribozymes are accurate enough to replicate their own sequences Eigen himself proposed a different solution to his paradox, and the next section will furnish a description of his solution 4 The hypercycle Eigen s paradox presents a serious obstacle to the acquisition of translational function by a lone self-replicating molecule However, we have already shown that no self-replicating molecule can completely outcompete other molecules that belong to the same quasispecies, such that it is unreasonable to picture a replicator evolving in isolation We assumed earlier that a quasispecies derives its fitness from the autocatalytic abilities of its isolated constituents, but it is natural to suppose that the related sequences making up a quasispecies could evolve cooperative behaviors in response to kin selection Experiments where digital organisms compete for computer processor space have shown that the fitness of a quasispecies can depend strongly on the fitness of the mutants it harbors; a 2001 study conducted at the Caltech Digital Life Laboratory reported that quasispecies with flat fitness peaks can outcompete other quasispecies centered around higher but narrower fitness peaks [35], where a few sequences are extremely fit but their close neighbors are considerably less fit Cooperation among members of a quasispecies could raise the average fitness of that quasispecies even more effectively than can fitness peak flatness, partly because a cooperative Page 10

11 of free-living molecules can survive replication at higher error rates than can a single molecule containing the same amount of information We can convince ourselves of this by a thought experiment, in which sequences A, B, and C form a molecular cooperative and replicate at an error rate that is below the minimum threshold at which each sequence can replicate individually, but is above the threshold at which the concatenation ABC could replicate In a pool where A, B, and C are joined to form one molecule, the abundance of perfect ABC master sequences will decline after each replication cycle, even as the abundance of sequences containing a perfect copy of A increases We can explain this by noting that the master sequence never makes up more than a few percent of a population replicating near the error threshold; even if a replication cycle increases the abundance of A from 5% to 6% and increases the abundance of B from 5% to 6%, it is unlikely that the perfect copies of A will be joined to the perfect copies of B, meaning that the perfect copies of ABC will decline and disappear In contrast, a population where A, B, and C are free-living and perfect copies of these sequences can find one another in solution will see the abundance of ABC molecular cooperatives increase from 5% to 6%, and the sequence information will survive Manfred Eigen and Peter Schuster explored different molecular cooperative structures using dynamical systems theory, varying the ways that different molecular species catalyze one another s replication They found that some catalytic networks dissolved as key molecular species went extinct, while others stabilized and prospered [7, 8, 9] In [8], Eigen and Schuster argue that the only stable autocatalytic network is an unbranched circular network that they call a hypercycle, and in [9], they describe a specific hypercycle that could account for the origin of translation in an RNA quasispecies The simplest natural hypercyles occur because of the fact that real nucleic acids strands are not autocatalytic The only truly autocatalytic single-stranded RNA molecules are palindromes; all other single-stranded RNA molecules template the replication of complementary sequences, and these complementary sequences return the favor Most modern catalytic RNA molecules are single strands that fold up into clover-leaf shapes with diverse tertiary structures, and such molecules must propagate themselves within catalytic cycles where complementary polymers catalyze one another s replication Eigen envisioned the hypercycle as a generalization of this process, with n molecular species E 1,, E n interacting such that E k catalyzes the formation of E k+1 for k < n and E n catalyzes the formation of E 1 A collection of ten RNA 100-mers organized into a hypercycle could theoretically replicate without the help of an optimally adapted protein polymerase, yet contains enough information to support a translation system We saw in the previous section that a dominant quasispecies outcompetes all other related and unrelated replicators in its environment This selective pressure makes it unlikely for a hypercycle to evolve from out of catalytic interactions among members of different quasispecies One might object that the information contained in ten RNA 100-mers belonging to the same quasispecies is less than the information content of a typical RNA 1000-mer, since molecules belonging to the same quasispecies have many residues in common, such that the Shannon information content of their concatenation is not maximal [32] However, the RNA sequences of modern translation apparatuses do not have maximal Shannon information content either Besides the ribosome that attaches a new amino acid to the end of a growing polypeptide, our cells express twenty transfer RNAs (trnas) that link free amino acids to the RNA codons that call for them, plus twenty aminoacyl-trna synthetases that attach amino acids to trnas according to the genetic code Two modern trnas are similar enough to be well-represented in the same quasispecies, as are two modern synthetases Eigen and Schuster postulate that the earliest translation apparatus could do without ribosomes and synthetases by substituting non-specific inorganic catalysts and modifying the structure of a trna [9], leaving an army of trnas that differ only in a few amino acidspecific residues and could easily inhabit one quasispecies Phylogenetic analysis of modern trna sequences suggest that they could have diverged from a common ancestor within a quasispecies framework [11, 12] In addition, a computer simulation of RNA evolution that assigns a selective advantage to rows of stable internal base pairs has a very high probability Page 11

12 Figure 2: In this simple hypercyclic translation apparatus (n = 4) [9], I 1,, I 4 are trnalike molecules that double as templates for the peptide catalysts E 1,, E 4 When E i is most efficient at catalyzing the replication of I i+1 and E 4 is most efficient at catalyzing the replication of I 1, all molecular species survive indefinitely, stabilizing at finite equilibrium concentrations If we complicate this hypercycle by introducing additional catalytic dependencies such as E 4 I 2, then as long as these dependencies facilitate reactions with weaker rate constants than the reactions depicted above, they will alter the position of equilibrium slightly without jeopardizing the stable coexistence of all replicators [9] of producing a clover leaf structure that resembles trna [5] These pieces of evidence add support to the idea that trna was the first part of the translation apparatus to evolve Eigen and Schuster assume that the first translation systems evolved when a set of replicators I 1,, I n, which belong to a single successful quasispecies, began to catalyze peptide synthesis by acting as trnas With these trnas present in solution, the polynucleotides I k could template the formation of polypeptides E k, and these polypeptides would amplify the production of the I k s When the system can do without synthetases and E i is a replicase specialized for the polymer x i+1 δnin, we get a hypercyclic structure (see figure 2) In theory, it is as plausible that the E k s will act as synthetases as it is plausible that they will act as replicases Eigen and Schuster propose two alternate causal networks where the products of translation function in this way (see figure 3) The arrows in these schematics represent causal links that show up in the set of differential equations that govern replicator proliferation as in Section 2 The diagrams only lack causal arrows between the I k replicators and the I k E k translation process because of an implicit assumption that templates, and not trnas, are the limiting reagents in the translation process This assumption will be valid at the beginning of the translation process because all I k s are abundantly produced in the quasispecies distribution, but it will stop being valid if any concentration ratio I k /I j increases unboundedly as the dynamical system evolves For this reason, a viable translation system is one where the participant concentrations I 1,, I n approach a stable equilibrium where all species coexist and the assumption of abundant trnas remains valid as the original quasispecies succumbs to competition and disappears The central project of [8] is to show that a simple, unbranched hypercycle is the only catalytic network that achieves such a stable equilibrium over time, and we now proceed to discuss the mathematics of this argument As before, we will start with polynucleotide replicators x 1,, x n and assume that the total concentration of polynucleotides c = n x n remains constant New polynucleotides enter the system via fitness-dependent replication as old ones leave by an outflow φ The rate φ i at which x i flows out of the system is dependent only on its abundance: φ i = x i c φ Page 12

13 Figure 3: These hypothetical primitive translation networks are heavily branched, meaning that they are not stable hypercycles [9] In the network on the left, peptide E 0 acts as an RNA polymerase, while all of the other catalytic peptides act as trna synthetases The collection of all four synthetases is required for translation to proceed, and this collection is depicted as a separate species F tr In the network on the right, all four peptides double as polymerases and synthetases However, the system is doomed to instability by the fact that E i replicates its parent polynucleotide I i, leading to excessive competition among the individual replicators When the x i are evolving as part of a simple quasispecies, we assume that that they obey linear growth equations x i = W i x i + w ik x k φ i k i In contrast, the advent of translation provides new ways for the x i s to catalyze one another s production, thereby introducing non-linearities into the equations governing system growth To reflect this fact, we will begin writing replicator functions in the general form The condition x i = Γ i ( x) φ i ( ċ = d n ) x i = 0, dt known as constant organization, means that outflow exactly balances replicator productivity, such that φ = Γ i ( x) and x i = Γ i ( x) x i c Γ j ( x) We have already seen that Γ i ( x) = W i x i when dealing with simple quasispecies In the simple hypercycle pictured above, the rate equations will include more nonlinearities, with j=1 Γ 1 ( x) = k 1 x 1 + k 1x n x 1 Γ i ( x) = k i x i + k ix i 1 x i 2 i n Page 13

14 for some rate constants k 1,, k n, k 1,, k n In general, it is difficult to solve nonlinear differential equations exactly However, we are less interested in describing the trajectory that a primitive translation system follows toward equilibrium than in deducing whether it ends up at an equilibrium where all catalytic species coexist Therefore, we will analyze each chemical system by fixing the total concentration of replicators and finding the critical points of the growth equations over the n-simplex whose points span all possible concentration ratios of x 1,, x n, then determining which of these critical points represent stable equilibria A critical point will represent a stable equilibrium if it is an attracting point, such that nearby states converge to it with time, and if it is in the interior of the concentration simplex, where all replicator concentrations are positive When the growth equations are homogeneous in the x i variables, the identities of their critical points do not depend upon the total concentration c [8] However, this independence does not hold in general, and so there may be a restricted range of concentrations at which the interior of the simplex contains one or more fixed points After locating the critical points at which x = 0, we can verify that some of these critical points represent equilibria using Lyapunov s second theorem on stability: Lemma 4 (Lyapunov s Second Theorem on Stability, [25]) Suppose that x = f( x) is an n-dimensional dynamical system and that x 0 is a critical point, such that x0 = 0 If there exist a function V ( x) and a neighborhood U x in R n for which V ( x 0 ) = 0, V ( x) > 0 for all x U, and ( ) V dxj V ( x) = < 0 x U, x j dt j=1 then x 0 is a stable equilibrium point, such that every dynamical trajectory passing sufficiently close to x 0 will move toward x 0 as t approaches infinity When n 4, Lyapunov s Second Theorem can be used to show that the simple, symmetric hypercycle with growth equations x 1 = k 1 x 1 + k 1x n x 1 x 1 c x i = k i x i + k ix i 1 x i x i c ( k 1 x 1 + k 1x n x 1 + k 1 x 1 + k 1x n x 1 + ) (k i x i + k ix i 1 x i ) i=2 j=2 ( kj x j + k jx ) j 1 x j 2 i n has a stable equilibrium point at which all species coexist [8] Computer approximations suggest that the same is true for larger values of n and for non-symmetric hypercycles, which do not obey the convenient constraints k i = c = 1 and k i = k j {0, 1} for all i, j {1,, n} [8] We can see that x 0 = (1/n,, 1/n) is a critical point of the dynamical system that describes a symmetric hypercycle, verifying easily that all relevant partial derivatives vanish In addition, the function vanishes at x 0 We compute that V ( x) = 1 n n x 1 x n Page 14

15 dv dt = x 1 x n x i x 1 x n i=2 x i ( x i = x 2 x n k 1 x 1 + k 1x n x 1 x 1 c k i x i + k ix i 1 x i x i c = x 1 x n k i + k ix i 1+nδi1 n c ( k 1 x 1 + k 1x n x 1 + k 1 x 1 + k 1x n x 1 + )) (k i x i + k ix i 1 x i ) i=2 ( kj x j + k jx ) j 1 x j j=2 k j x j + k jx j x j 1+nδ1j j=1 When we make the substitutions k i = c = 1 and k i = k j {0, 1} and use the fact that n x i = c = 1, we find that dv dt ( n ) = x 1 x n k i k i x i + x i n k i x i x i 1+nδi1 = x 1 x n (1 n ) x i x i 1+nδi1 When n 4, we can verify that dv/dt < 0 at all interior points of the concentration simplex except for the central fixed point, proving that this point is stable For n 5, it is not possible to prove this stability by Lyapunov s method, but numerical integration suggests that stability still holds [8] Looking back at Figure 3, we see two hypothetical translation mechanisms that are not simple hypercycles, but branched networks A computation very similar to the previous one locates no internal attracting points within the representation of a branched network on a replicator concentration simplex Therefore, branched networks are inherently unstable, as are hypercycles with parasitic couplings [8] This suggests that the earliest translation system can be represented by a cyclic set of catalytic linkages, which motivates Eigen and Schuster to favor the synthetase-free model featured in Figure 2 This model has its problems, considering that there is no biochemical example of a trna that can link itself to a specific amino acid without enzymatic help Any biochemical manifestation of this process would have a nonzero error rate, which would compound with ordinary transcription and translation errors One of the most important consequences of the mathematical analysis in [8] is the knowledge that these obstacles affect the general viability of catalytic cooperation among replicators; we have proof that we should not waste our time studying branched networks that eliminate the problems of a synthetase-free scenario, seeing that network stability constraints make these scenarios unlikely to account for the origin of translation Instead, we must focus on the more obviously problematic cyclic networks Serious problems with the biochemistry of cyclic translation reactions also magnify the importance of completely different solutions to Eigen s paradox, which would allow for protein translation to emerge later in the history of life among chemical species that are subject to less constrained dynamic behavior One alternative to hypercyclic organization is a relaxation of the error threshold that is supported by experimental evidence that replication slows during the production of mutant sequences [21, 22] If this stalling dampens the production of mutants effectively enough, then hypercyclic organization may be less crucial for the advent of long genomes than Eigen and Schuster originally supposed One salient feature of protein translation is its reliance on a near-universal genetic code Scientists looking for patterns in the genetic code have noted that the simplest amino acids are specified by the codons that would have replicated most easily without complex enzymes [9], and others have noted that the genetic code maximizes the likelihood that a point mutation will induce the substitution of an amino acid that is chemically similar to the wildtype residue [15] At the same time, critics of such pattern-finding efforts wonder how much Page 15

16 of the code is a frozen accident that persisted because of evolutionary inertia [4] There is mathematical evidence that the selection of a hypercycle is a once-forever decision, partly because hypercycles resist branching and parasitism [8] A hypercycle s nonlinear growth rate will help it easily outcompete any independently replicating quasispecies, eliminating any precursors to competing hypercycles from its environment Despite our lack of definitive knowledge about the earliest translation system, it was almost certainly complex enough to grow at a nonlinear rate, making it subject to once-forever selection dynamics The systems that were analyzed in Sections 3 and 4 revolve around a scenario for the emergence of translation and high-fidelity transcription in a pool of polymers that are already capable of crude self-replication Much less theory has focused upon the de novo emergence of self-replication, perhaps because there is no modern biological process that seems to recapitulate this event in the way that translation ribozymes hearken back to an RNA world A 2008 paper by Martin Nowak and Hisashi Ohtsuki tackles this problem via a model where natural selection precedes replication [28] In the next section, we present this work as a possible foundation for Eigen and Schuster s older origins of life scenarios 5 Prevolutionary dynamics and the origin of replication Like classical quasispecies theory, the process that Nowak and Ohtsuki have dubbed prevolution is described by a set of ordinary differential equations whose variables x i represent polynucleotide concentrations [28] As before, we assume that the solution contains a ready supply of activated nucleotides, and for simplicity we consider only two bases denoted 0 and 1 In Eigen s theory, the activated nucleotides 0* and 1* react with polynucleotides via a black-box transcription mechanism such that the sequences x i beget copies of themselves In contrast, prevolutionary dynamics allows for activated nucleotides to tack themselves onto any polynucleotide strings they encounter, transforming the strand x i into the strands x i0 and x i1 Activated nucleotides can also become deactivated hydrolytically, begetting the monomeric strands x 0 and x 1 To compensate for excess production, the polymer x i decays at the rate dx i Letting x i denote the species obtained by removing the terminal nucleotide from x i and a i denote the rate at which x i is transformed into x i, we obtain the growth equations x i = a i x i (d + a i0 + a i1 )x i Since activated nucleotides become deactivated by reacting with superabundant water molecules, we can assume that x 0 = x 1 = 1 Given that prevolutionary dynamics obey a set of linear differential equations with constant coefficients, they are much simpler to analyze mathematically than either primitive replicator quasispecies or catalytic networks Letting x = (x 0 = x 1, x 0, x 1, x 00, x 01, x 10, x 11, ) be the infinite vector whose entries are the finite binary strings, we can write ẋ = Ax for the constant matrix A = 1 a 0 d a 00 a 01 a 1 d a 10 a 11 a 00 d a 000 a 001 This implies that the solutions of the prevolutionary dynamics equations have the form x = exp(at)x 0, where x 0 is a constant vector of initial conditions A is clearly diagonalizable with real eigenvalues, the largest eigenvalue corresponding to an eigenvector that is a stationary, limiting distribution of polynucleotides The easiest way to calculate this distribution is to solve the equation system ẋ = 0, which holds only at equilibrium [28] Doing so, we obtain the equations x i a i = := b i x i d + a i0 + a i1 Page 16